\(\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx\) [408]

Optimal result

Integrand size = 30, antiderivative size = 488 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\frac {15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt {a+i a \tan (c+d x)}}-\frac {15 i a^{7/2} e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {15 i a^{7/2} e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {15 i a^{7/2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)} \left (\sqrt {a}+\cos (c+d x) \left (\sqrt {a}-i \sqrt {a} \tan (c+d x)\right )\right )}\right ) \sec (c+d x)}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {3 i a^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{4 d}+\frac {i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d} \] Output:

15/8*I*a^3*(e*sec(d*x+c))^(3/2)/d/(a+I*a*tan(d*x+c))^(1/2)-15/16*I*a^(7/2) 
*e^(3/2)*arctan(1-2^(1/2)*e^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/a^(1/2)/(e*sec( 
d*x+c))^(1/2))*sec(d*x+c)*2^(1/2)/d/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d* 
x+c))^(1/2)+15/16*I*a^(7/2)*e^(3/2)*arctan(1+2^(1/2)*e^(1/2)*(a-I*a*tan(d* 
x+c))^(1/2)/a^(1/2)/(e*sec(d*x+c))^(1/2))*sec(d*x+c)*2^(1/2)/d/(a-I*a*tan( 
d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)-15/16*I*a^(7/2)*e^(3/2)*arctanh(2^( 
1/2)*e^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/(e*sec(d*x+c))^(1/2)/(a^(1/2)+cos(d* 
x+c)*(a^(1/2)-I*a^(1/2)*tan(d*x+c))))*sec(d*x+c)*2^(1/2)/d/(a-I*a*tan(d*x+ 
c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)+3/4*I*a^2*(e*sec(d*x+c))^(3/2)*(a+I*a*t 
an(d*x+c))^(1/2)/d+1/3*I*a*(e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(3/2)/d
 

Mathematica [A] (warning: unable to verify)

Time = 2.81 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.79 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\frac {\cos ^4(c+d x) (e \sec (c+d x))^{3/2} \left (\frac {1}{6} \sec ^3(c+d x) (63+79 \cos (2 (c+d x))+34 i \sin (2 (c+d x))) (i \cos (3 c+d x)+\sin (3 c+d x))+\frac {15 \left (\text {arctanh}\left (\frac {\sqrt {1+i \cos (c)-\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \sqrt {-1-i \cos (c)-\sin (c)} \sqrt {1+i \cos (c)-\sin (c)}-\text {arctanh}\left (\frac {\sqrt {1-i \cos (c)+\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1-i \cos (c)-\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \sqrt {1-i \cos (c)+\sin (c)} \sqrt {-1+i \cos (c)+\sin (c)}\right ) (\cos (3 c)-i \sin (3 c)) \sqrt {i+\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1-i \cos (c)-\sin (c)} \sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}\right ) (a+i a \tan (c+d x))^{5/2}}{8 d (\cos (d x)+i \sin (d x))^2} \] Input:

Integrate[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]
 

Output:

(Cos[c + d*x]^4*(e*Sec[c + d*x])^(3/2)*((Sec[c + d*x]^3*(63 + 79*Cos[2*(c 
+ d*x)] + (34*I)*Sin[2*(c + d*x)])*(I*Cos[3*c + d*x] + Sin[3*c + d*x]))/6 
+ (15*(ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[ 
-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[ 
c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqr 
t[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]]) 
]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*(Cos[3*c] - I* 
Sin[3*c])*Sqrt[I + Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[-1 + 
I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]]))*(a + I*a*Tan[c + d*x])^(5/2))/ 
(8*d*(Cos[d*x] + I*Sin[d*x])^2)
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.03, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3979, 3042, 3979, 3042, 3979, 3042, 3980, 3042, 3976, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2}dx\)

\(\Big \downarrow \) 3979

\(\displaystyle \frac {3}{2} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^{3/2}dx+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^{3/2}dx+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3979

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3979

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3980

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 3976

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \int \frac {\cos (c+d x) (a-i a \tan (c+d x))}{e \left (a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\int \frac {a+\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}+\frac {\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} e}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {a} e}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\)

Input:

Int[(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]
 

Output:

((I/3)*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/d + (3*a*((( 
I/2)*a*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/d + (5*a*((I*a*( 
e*Sec[c + d*x])^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + ((2*I)*a^2*e^3*((- 
(ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*S 
ec[c + d*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sq 
rt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a] 
*Sqrt[e]))/(2*e) - (-1/2*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan 
[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]/(S 
qrt[2]*Sqrt[a]*Sqrt[e]) + Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Ta 
n[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]/( 
2*Sqrt[2]*Sqrt[a]*Sqrt[e]))/(2*e))*Sec[c + d*x])/(d*Sqrt[a - I*a*Tan[c + d 
*x]]*Sqrt[a + I*a*Tan[c + d*x]])))/4))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3976

Int[Sqrt[(d_.)*sec[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.) 
*(x_)]], x_Symbol] :> Simp[-4*b*(d^2/f)   Subst[Int[x^2/(a^2 + d^2*x^4), x] 
, x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]], x] /; FreeQ[{a, b, d, 
e, f}, x] && EqQ[a^2 + b^2, 0]
 

rule 3979

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n 
 - 1)/(f*(m + n - 1))), x] + Simp[a*((m + 2*n - 2)/(m + n - 1))   Int[(d*Se 
c[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, 
 m}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ 
[2*m, 2*n]
 

rule 3980

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(3/2)/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_ 
.)*(x_)]], x_Symbol] :> Simp[d*(Sec[e + f*x]/(Sqrt[a - b*Tan[e + f*x]]*Sqrt 
[a + b*Tan[e + f*x]]))   Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*x]], 
 x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
 

Maple [A] (verified)

Time = 7.69 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {e \sqrt {e \sec \left (d x +c \right )}\, a^{2} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (-45 i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )+45 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-45 i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )-1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-45 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )-1}{2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-2 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (8 \sec \left (d x +c \right ) \tan \left (d x +c \right )-45 \sin \left (d x +c \right )+34 \tan \left (d x +c \right )\right )-2 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (79+45 \cos \left (d x +c \right )+26 \sec \left (d x +c \right )-8 \sec \left (d x +c \right )^{2}\right )\right )}{48 d \left (-\sin \left (d x +c \right )+i \cos \left (d x +c \right )+i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\)	\(316\)

Input:

int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/48/d*e*(e*sec(d*x+c))^(1/2)*a^2*(a*(1+I*tan(d*x+c)))^(1/2)/(-sin(d*x+c)+ 
I*cos(d*x+c)+I)/(1/(cos(d*x+c)+1))^(1/2)*(-45*I*cos(d*x+c)*arctanh(1/2/(1/ 
(cos(d*x+c)+1))^(1/2)*(cot(d*x+c)-csc(d*x+c)+1))+45*cos(d*x+c)*arctanh(1/2 
/(1/(cos(d*x+c)+1))^(1/2)*(cot(d*x+c)-csc(d*x+c)+1))-45*I*cos(d*x+c)*arcta 
nh(1/2*(cot(d*x+c)-csc(d*x+c)-1)/(1/(cos(d*x+c)+1))^(1/2))-45*cos(d*x+c)*a 
rctanh(1/2*(cot(d*x+c)-csc(d*x+c)-1)/(1/(cos(d*x+c)+1))^(1/2))-2*I*(1/(cos 
(d*x+c)+1))^(1/2)*(8*sec(d*x+c)*tan(d*x+c)-45*sin(d*x+c)+34*tan(d*x+c))-2* 
(1/(cos(d*x+c)+1))^(1/2)*(79+45*cos(d*x+c)+26*sec(d*x+c)-8*sec(d*x+c)^2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.30 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fric 
as")
 

Output:

1/12*((113*I*a^2*e*e^(4*I*d*x + 4*I*c) + 126*I*a^2*e*e^(2*I*d*x + 2*I*c) + 
 45*I*a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) 
 + 1))*e^(1/2*I*d*x + 1/2*I*c) + 6*sqrt(225/64*I*a^5*e^3/d^2)*(d*e^(4*I*d* 
x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/15*(15*(a^2*e*e^(2*I*d*x + 
 2*I*c) + a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2* 
I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 8*I*sqrt(225/64*I*a^5*e^3/d^2)*d)/(a^ 
2*e)) - 6*sqrt(225/64*I*a^5*e^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d 
*x + 2*I*c) + d)*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/( 
e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 
 1/2*I*c) - 8*I*sqrt(225/64*I*a^5*e^3/d^2)*d)/(a^2*e)) + 6*sqrt(-225/64*I* 
a^5*e^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/1 
5*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1) 
)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 8*I*sqrt(-22 
5/64*I*a^5*e^3/d^2)*d)/(a^2*e)) - 6*sqrt(-225/64*I*a^5*e^3/d^2)*(d*e^(4*I* 
d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/15*(15*(a^2*e*e^(2*I*d*x 
 + 2*I*c) + a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 
2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) - 8*I*sqrt(-225/64*I*a^5*e^3/d^2)*d)/ 
(a^2*e)))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [F(-1)]

Timed out. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((e*sec(d*x+c))**(3/2)*(a+I*a*tan(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3005 vs. \(2 (362) = 724\).

Time = 0.47 (sec) , antiderivative size = 3005, normalized size of antiderivative = 6.16 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxi 
ma")
 

Output:

192*(1808*a^2*e*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 201 
6*a^2*e*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 720*a^2*e*c 
os(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1808*I*a^2*e*sin(9/4 
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2016*I*a^2*e*sin(5/4*arcta 
n2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 720*I*a^2*e*sin(1/4*arctan2(sin( 
2*d*x + 2*c), cos(2*d*x + 2*c))) - 90*(sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 3* 
sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + I*sqrt 
(2)*a^2*e*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 3*I*sqrt 
(2)*a^2*e*sin(2*d*x + 2*c) + sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arctan 
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, sqrt(2)*sin(1/4*arctan2(sin(2* 
d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 90*(sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 
 3*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + I*s 
qrt(2)*a^2*e*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 3*I*s 
qrt(2)*a^2*e*sin(2*d*x + 2*c) + sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arc 
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, -sqrt(2)*sin(1/4*arctan2(si 
n(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 90*(sqrt(2)*a^2*e*cos(6*d*x + 6* 
c) + 3*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + 
 I*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 3 
*I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4 
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1, sqrt(2)*sin(1/4*arct...
 

Giac [F(-2)]

Exception generated. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac 
")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \] Input:

int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^(5/2),x)
 

Output:

int((e/cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i)^(5/2), x)
 

Reduce [F]

\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\frac {\sqrt {e}\, \sqrt {a}\, a^{2} e \left (-2 \sqrt {\sec \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right ) i -\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}d x \right ) d +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )d x \right ) d i \right )}{d} \] Input:

int((e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(5/2),x)
 

Output:

(sqrt(e)*sqrt(a)*a**2*e*( - 2*sqrt(sec(c + d*x))*sqrt(tan(c + d*x)*i + 1)* 
sec(c + d*x)*i - int(sqrt(sec(c + d*x))*sqrt(tan(c + d*x)*i + 1)*sec(c + d 
*x)*tan(c + d*x)**2,x)*d + 6*int(sqrt(sec(c + d*x))*sqrt(tan(c + d*x)*i + 
1)*sec(c + d*x)*tan(c + d*x),x)*d*i))/d